Page 149 - Contributed Paper Session (CPS) - Volume 1

CPS1216 Teppei O.

̃

̂
̃
Let ∑ () = ∑( −1 , −1 , ), , = ( ), , = +1+ −1 − + −1 and

= (( 1 ) , … , ( , ) ) .
,

For a matrix A, we denote its (, )-element by [] . Then roughly speaking,

we obtain approximation:

E[ , , | s m-1 ] ≈ [∑ s m-1 , †] |I , | + [ , ] ,

,∗

E[ , , | s m-1 ] ≈ [∑ s m-1 , †] |I | ∩ , |

,

for ≠ . Therefore, by setting
1
Υ
[] diag ((| , |) ) ⋯ [] {| 1 , ∩ , |}
11
1Υ
1 1,
(, ) = ⋮ ⋱ ⋮ + ( ⋱ )

1
Υ
[] Υ1 {| , ∩ , |} ⋯ [] ΥΥ diag ((| , |) ) ,

( )

for a × matrix B and = ( , … , ), and (, ) = (∑ (), ), we

1

define a quasi-log-likelihood function

1
−1

(, ) = − ∑ ( (, ) + log det (, )).

2
=2

The function contains two parameters: the first one is , which is the

parameter for estimator of ∑ and is of our interest. The second parameter is
, which is the parameter for noise variance. Though we consider a
simultaneous maximization of and , we accept beforehand estimation of
so that we apply our results to the case of non-Gaussian noise.
[V] There exist estimators {̂ } of such that ̂ ≥ 0 almost surely and

∗
∈ℕ
⁄
{ 1 2 (̂ − )} is tight.

∗
∈ℕ
Such ̂ can be easily obtained. For example, let ̂ = (̂ , … , ̂ and

1

−1 2

̂ , = (2J , ) ∑( , )
,

4
Then ̂ satisfies [V] if { J −1 } ∈ℕ is tight and sup ,, [(∈ , ) ] < ∞.
,

We fix ̂ which satisfies [V] . We define a maximum-likelihood-type

estimator
̂ =argmaxσHn(, ̂ ).

is constructed based on local Gaussian approximation of . This

,
approximation seems valid only when observation noise ∈ , follows a normal

138 | I S I W S C 2 0 1 9